Published for SISSA by
Springer
Received: April 22,
Revised: August 10,
Accepted: September 2,
Published: September 25,
2014
2014
2014
2014
Seungwon Baek,a P. Ko,a Hiroshi Okadaa and Eibun Senahaa,b
a
School of Physics, KIAS,
85 Hoegiro Dongdaemun-gu, Seoul 130-722, Korea
b
Department of Physics, Nagoya University,
Furo-cho, Chikusa-ku, Nagoya 464-8602, Japan
E-mail: [email protected], [email protected], [email protected],
[email protected]
Abstract: We extend the Zee-Babu model for the neutrino masses and mixings by first
incorporating a scalar dark matter X with Z2 symmetry and then X and a dark scalar ϕ
with global U(1) symmetry. In the latter scenario the singly and doubly charged scalars
that are new in the Zee-Babu model can explain the large annihilation cross section of a
dark matter pair into two photons as hinted by the recent analysis of the Fermi γ-ray space
telescope data. These new scalars can also enhance the B(H → γγ), as the recent LHC
results may suggest. The dark matter relic density can be explained. The direct detection
rate of the dark matter is predicted to be about one order of magnitude down from the
current experimental bound in the first scenario.
Keywords: Phenomenological Models
ArXiv ePrint: 1209.1685
c The Authors.
Open Access, Article funded by SCOAP3 .
doi:10.1007/JHEP09(2014)153
JHEP09(2014)153
Can Zee-Babu model implemented with scalar dark
matter explain both Fermi-LAT 130 GeV γ-ray excess
and neutrino physics?
Contents
1
2 The
2.1
2.2
2.3
2.4
3
3
5
6
7
Z2 model
Constraints on the potential
XX → γγ and Fermi-LAT 130 GeV γ-ray excess
Thermal relic density and direct detection rate
H → γγ, Zγ
3 Spontaneously broken U(1)B−L model
3.1 XR XR → γγ and Fermi-LAT 130 GeV γ-ray excess in U(1)B−L model
3.2 Relic density in U(1)B−L model
3.3 Other cosmological implications of U(1)B−L model: ∆Neff , topological defects, self-interacting dark matter
3.4 Implications for neutrino physics in both Z2 and U(1)B−L model
9
12
14
4 Conclusions
20
A One-loop β functions of the quartic couplings
21
B The annihilation cross section of XR XR → αα
22
1
18
20
Introduction
Although it is well known that the dark matter (DM) constitutes about 27% of the total
mass density of the universe, i.e. ΩDM h2 = 0.1199 ± 0.0027 [1], its existence has only been
inferred from the gravitational interaction. And its nature is still unknown. If the DM
is weakly interacting massive particle (WIMP), it may reveal itself via non-gravitational
interactions, for example, by pair annihilation into ordinary standard model (SM) particles
including photon [2]. In this case, the DM relic abundance is roughly related to the pair
annihilation cross section at freezeout, hσvith , as
ΩDM h2 =
3 × 10−27 cm3 /s
.
hσvith
(1.1)
Refs. [13–16], more recently ref. [17] but with less significance, claim that the Fermi γ-ray
space telescope may have seen excess of the photons with Eγ ∼ 130 GeV from the center of
the Milky Way compared with the background. Interpreting its origin as the annihilation
of a pair of DM particles, they could obtain the annihilation cross section to be about 4%
of that at freezeout:
hσviγγ ≈ 0.04hσvith ≈ 0.04 pb ≈ 1.2 × 10−27 cm3 /s.
–1–
(1.2)
JHEP09(2014)153
1 Introduction
Since DM is electrically neutral, the pair annihilation process into photons occurs through
loop-induced diagrams. Naively we expect
α 2
hσviγγ
em
=
∼ 10−5 .
hσvith
π
(1.3)
–2–
JHEP09(2014)153
So the observed value in (1.2) is rather large, and we may need new electrically charged particles running inside the loop beyond the SM. Many new physics scenarios were speculated
within various CDM models by this observation [18–43].
The so-called ‘Zee-Babu model’ [44–46] provides new charged scalars, h+ , k ++ at electroweak scale, in addition to the SM particles. These new charged scalars carry two units
of lepton number and can generate Majorana neutrino masses via two-loop diagrams. The
diagrams are finite and calculable. The neutrino masses are naturally small without the
need to introduce the right-handed neutrinos for seesaw mechanism. One of the neutrinos
is predicted to be massless in this model. Both normal and inverse hierarchical pattern
of neutrino masses are allowed. The observed mixing pattern can also be accommodated.
The model parameters are strongly constrained by the neutrino mass and mixing data, the
radiative muon decay, µ → eγ, and τ → 3µ decay [47].
It would be very interesting to see if the new charged particles in the Zee-Babu model
can participate in some other processes in a sector independent of neutrinos. In the first
part of this paper, we minimally extend the Zee-Babu model to incorporate the DM. In
the later part we will consider more extended model with global U(1)B−L symmetry and
an additional scalar which breaks the global symmetry [50].
In the first scenario, we introduce a real scalar dark matter X with a discrete Z2
symmetry under which the dark matter transforms as X → −X in order to guarantee
its stability. The renormalizable interactions between the scalar DM X with the Higgs
field and the Zee-Babu scalar fields provide a Higgs portal between the SM sector and the
DM. We show that the Zee-Babu scalars and their interactions with the DM particle can
explain the DM relic density. The branching ratio of Higgs to two photons, B(H → γγ),
can also be enhanced as implied by the recent LHC results [63–65]. Although the charged
Zee-Babu scalars enhance the XX → γγ process, it turns out the the current experimental
constraints on their masses do not allow the annihilation cross section to reach (1.2).
In the extended scenario, we consider a complex scalar dark matter X and a dark scalar
ϕ [50]. The global U(1) symmetry of the original Lagrangian is broken down to Z2 by ϕ getting a vacuum expectation value (vev). We show that both the dark matter relic abundance
and the Fermi-LAT gamma-ray line signal can be accommodated via two mechanisms.
This paper is organized as follows. In section 2, we define our model by including the
scalar DM in the Zee-Babu model, and consider theoretical constraints on the scalar potential. Then we study various DM phenomenology. We calculate the dark matter relic density
and the annihilation cross section hσviγγ in our model. We also predict the cross section
for the DM and proton scattering and the branching ratio for the Higgs decay into two
photons, B(H → γγ). And we consider the implication on the neutrino sector. In section 3,
we consider the DM phenomenology in the extended model. We conclude in section 4.
2
The Z2 model
We implement the Zee-Babu model for radiative generation of neutrino masses and mixings, by including a real scalar DM X with Z2 symmetry X → −X. All the possible
renormalizable interactions involving the scalar fields are given by
L = LBabu + LHiggs+DM
LBabu =
j
Ti
fab laL
ClbL
ǫij h+
+
(2.1)
′
T
hab laR
ClbR k ++
+ h.c.
(2.3)
Note that our model is similar to the model proposed by J. Cline [19]. However we included
the interaction between the new charged scalar and the SM leptons that are allowed by
gauge symmetry, and thus the new charged scalar bosons are not stable and cause no
problem.
The original Zee-Babu model was focused on the neutrino physics, and the operators of
Higgs portal types were not discussed properly. It is clear that those Higgs portal operators
we include in the 2nd line of (2.3) can enhance H → γγ, without touching any other decay
rates of the SM Higgs boson, as long as h± and k ±± are heavy enough that the SM Higgs
decays into these new scalar bosons are kinematically forbidden.
2.1
Constraints on the potential
We require µ2X , µ2h and µ2k to be positive. Otherwise the imposed Z2 symmetry X → −X
or the electromagnetic U(1) symmetry could be spontaneously broken down. Since the
masses of X, k ++ and h+ have contributions from the electroweak symmetry breaking as
1
2
m2X = µ2X + λHX vH
,
2
1
2
m2h+ = µ2h + λHh vH
,
2
1
2
m2k++ = µ2k + λHk vH
,
2
(2.4)
we obtain the conditions on the quartic couplings
λHX <
2m2X
2 ,
vH
λHh <
2m2h+
2 ,
vH
λHk <
2m2k++
.
2
vH
(2.5)
We note that the above conditions are automatically satisfied if the couplings takes negative
values. In such a case, however, we also need to worry about the behavior of the Higgs
–3–
JHEP09(2014)153
1
−LHiggs+DM = −µ2H H † H + µ2X X 2 + µ2h h+ h− + µ2k k ++ k −−
2
+(µhk h− h− k ++ + h.c.)
1
+λH (H † H)2 + λX X 4 + λh (h+ h− )2 + λk (k ++ k −− )2
4
1
+ λHX H † HX 2 + λHh H † Hh+ h− + λHk H † Hk ++ k −−
2
1
1
+ λXh X 2 h+ h− + λXk X 2 k ++ k −− + λhk h+ h− k ++ k −− .
2
2
(2.2)
potential for large field values. For example, if we consider only the neutral Higgs field,
H,1 and the dark matter field, X, we get
1
1
1
λH H 4 + λX X 4 + λHX H 2 X 2 ,
4
4
4
!
!
1
2
1
H
λ
λ
H
HX
2
,
∼
H2 X2
1
4
λ
λX
X2
HX
2
V ∼
(2.6)
This means that even if λHX is negative, its absolute value should not be arbitrarily large
2 ≈ 0.13 (m ≈ 125 GeV) and λ is bounded from above so as not
because λH = m2H /2vH
H
X
to generate the Landau pole. For example, the renormalization group running equations
(RGEs) of λH , λX and λHX are given by
1 2
1
dλH
2
24λH + λHX + · · · ,
=
d log Q
16π 2
2
dλX
1
2
2
=
18λ
+
2λ
+ ··· ,
X
HX
d log Q
16π 2
λHX dλHX
=
6λ
+
3λ
+ ··· ,
(2.8)
H
X
d log Q
8π 2
where the dots represents other contributions which are not important in the discussion.
The complete forms of the β-functions of the quartic couplings are listed in appendix A.
The approximate solution for λX in (2.8) shows that the Landau pole is generated at
the scale Q = QEW exp (1/βH λX (QEW )) (βH = 18/16π 2 ). If we take the electroweak scale
value of the Higgs quartic coupling to be λX (QEW ) ∼ 5, the cut-off scale should be around
1 TeV. The general condition for the bounded-from-below potential for large field values is
that all the eigenvalues of the matrix


λH 12 λHX λHh λHk
 1λ

 2 HX λX λXh λXk 
(2.9)


 λHh λXh 4λh 2λhk 
λHk λXh 2λhk 4λk
should be positive.
In the following discussion, we require that all scalar quartic couplings (λi ) be perturbative up to some scale Q. To this end, we solve the one-loop RGEs of those quartic couplings
given in appendix A. For the moment, we do not include new Yukawa couplings defined in
eq. (2.2), and we adopt the criterion λi (Q) < 4π in this analysis. In figure 1, the perturbativity bounds are shown in the λXh(k) -λHh(k) plane. We take Q = 1, 3, 10 and 15 TeV, which
are denoted by the red curves from top to bottom. For other parameters, we fix λHh = λHk ,
1
We use the same notation with the Higgs doublet.
–4–
JHEP09(2014)153
for large field values of H and X. If the potential is to be bounded from below, every eigenvalue of the square matrix of the couplings in (2.6) should be positive, whose condition is
p
|λHX | < 4λH λX .
(2.7)
λXh = λXk , λhk = λHX = 0 and λX = λH (≃ 0.13). As explained above, a certain negative
value of λHk(h) may cause the instability of the Higgs potential. To avoid this, we set
λh = λ2Hh /(2λH ) and λk = λ2Hk /(2λH ) for λHk(h) < 0. For λHk(h) > 0, on the other hand,
λh = λk = λH is taken. As we see from the plot, λXk(h) ≃ 7 − 11 is possible if Q = 1 TeV.
The theoretical arguments (2.5) and (2.7) restrict λHX to lie roughly to the range,
(−1.6, 0.6). Similarly, we have λHh(k) . 0.7 for mh+ (k++ ) = 150 GeV.
2.2
XX → γγ and Fermi-LAT 130 GeV γ-ray excess
The annihilation cross section for XX → γγ is given by
P
|M|2
,
hσviγγ =
64πm2X
where the amplitude-squared summed over the photon polarization is
2 X
αem
2
|M| =
λXh A0 (τh+ ) + 4λXk A0 (τk++ )
2π 2 2 2
λHX vH
g
2
+
Q
N
A
(τ
)
+
A
(τ
)
1 W
t C 1/2 t
s − m2H + imH ΓH 2τW
2
+λHh A0 (τh+ ) + 4λHk A0 (τk++ ) ,
with τi = 4m2i /s(i = h+ , k ++ , t, W ). The loop functions are
A0 (τ ) = 1 − τ f (τ ),
–5–
(2.10)
(2.11)
JHEP09(2014)153
Figure 1. The perturbativity bounds, λi (Q) < 4π are shown. The each curve denotes Q = 1,
3, 10 and 15 TeV from top to bottom. We take λHh = λHk , λXh = λXk , λhk = λHX = 0 and
λX = λH (≃ 0.13). For the negative λHk , we set λh = λ2Hh /(2λH ) and λk = λ2Hk /(2λH ) while
λh = λk = λH for positive λHk .
H a L Λ HX =Λ Hh =Λ Hk =0
10
H b L Λ HX =Λ Hh =Λ Hk =0.33
10
2 ´10 - 27 cm 3  s
2 ´10 - 27 cm 3  s
8
8
1 ´10 - 27 cm 3  s
6
0.5 ´10
- 27
1 ´10 - 27 cm 3  s
Λ Xh
Λ Xh
6
3
cm  s
4
4
0.5 ´10 - 27 cm 3  s
0.2 ´10 - 27 cm 3  s
0.2 ´10 - 27 cm 3  s
2
0
0
100
110
120
130
140
150
100
110
m h + H GeVL
120
130
140
150
m h + H GeVL
Figure 2. Contour plot of hσviγγ = (2, 1, 0.5, 0.2) × 10−27 cm3 /s (from above) in (mh+ , λXh ) plane.
We set mX = 130 GeV, mH = 125 GeV, mk = 500 GeV and λXk = 5, λHX = λHh = λHk = 0(0.33)
in the left (right) panel.
h
i
A1/2 (τ ) = −2τ 1 + (1 − τ )f (τ ) ,
A1 (τ ) = 2 + 3τ + 3τ (2 − τ )f (τ ),
where

p
 arcsin2 1/τ , (τ ≥ 1)
i2
h
√
f (τ ) =
 − 14 log 1+√1−τ − iπ , (τ < 1).
1− 1−τ
(2.12)
(2.13)
Although the contribution of the doubly-charged Higgs k ++ to hσviγγ is 24 = 16 times
larger than that of the singly-charged Higgs h+ when their masses are similar to each other,
this option is ruled out by the recent LHC searches for the doubly-charged Higgs boson [66].
Depending on the decay channels, the 95% CL lower limit on the mass of the doubly-charged
Higgs boson is in the range, 204–459 GeV. To be conservative, we set mk++ = 500 GeV.
In figure 2, we show a contour plot for the annihilation cross section into two photons:
hσviγγ ≈ (2, 1, 0.5, 0.2) × 10−27 cm3 /s (from above) in the (mh+ , λXh ) plane. We set mX =
130 GeV, mH = 125 GeV, mk++ = 500 GeV and λXk = 5, λHX = λHh = λHk = 0 (0.33)
in the left (right) panel. We can see that by turning on the process, XX → H → γγ, with
λHX = 0.33 (right panel), we can reduce λXh to get hσviγγ = 1 × 10−27 cm3 /s to explain
the Fermi-LAT gamma-ray line signal, but not significantly enough to push the cut-off
scale much higher than the electroweak scale. As we will see in the following section, the
hσviγγ = 1 × 10−27 cm3 /s is not consistent with the current DM relic abundance.
2.3
Thermal relic density and direct detection rate
Contrary to J. Cline’s model [19], the DM relic density in our model is not necessarily correlated with the hσviγγ , since it is mainly determined by λHX for relatively heavy scalars
(& 150 GeV). In this case the main DM annihilation channels are XX → H → SM particles,
–6–
JHEP09(2014)153
2
mH=125 GeV, mX=130 GeV
0.05
ΛHX
0.04
0.03
0.01
-10
0
-5
5
10
ΛXh
Figure 3. The contour plot of ΩDM h2 = 0.1199 (red lines) and hσviγγ = 0.2 × 10−27 cm3 /s (black
lines) in the (λXh ,λHX ) plane for the choices mh+ = 150, 140, 130 GeV (solid, dashed, dotted
lines). For other parameters we set mX = 130 GeV, mH = 125 GeV, mk = 500 GeV, λXk = 5,
λHh = λHk = 0.5.
where the SM particles are W + W − , ZZ, bb̄, etc. As mh+ (k++ ) becomes comparable with
mX , the XX → h+ h− (k ++ k −− ) modes can open, even in cases mX < mh+ (mk++ ) due to
the kinetic energy of X at freeze-out time. This can be seen in figure 3, where we show the
contour plot of ΩDM h2 = 0.1199 (red lines) in the (λXh ,λHX ) plane for the choices mh+ =
150, 140, 130 GeV (shown in solid, dashed, dotted lines respectively). We fixed other parameters to be mX = 130 GeV, mH = 125 GeV, mk = 500 GeV, λXk = 5, λHh = λHk = 0.5.
For mh+ = 130 GeV, the annihilation mode XX → h+ h− dominates even for very small
coupling λXh (the red dotted line). The black vertical lines are the constant contour lines of
hσviγγ = 0.2×10−27 cm3 /s. We can see that the maximum value for the Fermi-LAT gammaray line signal which is consistent with the relic density is hσviγγ = 0.2 × 10−27 cm3 /s when
mh+ = 150 GeV. This cross section is smaller than the required value in (1.2) by factor 6.
Figure 4 shows the cross section of dark matter scattering off proton, σp , as a function
of λHX (red solid line) and σp = 1.8 × 10−9 pb line (black dashed line) above which is
excluded by LUX [67] at 90% C.L. This cross section is determined basically only by λHX
at tree level by the SM Higgs exchange, when we fix mX = 130 GeV. We can see that
λHX . 0.06 to satisfy the LUX upper bound.
2.4
H → γγ, Zγ
In this scenario the decay width of H → γγ, Zγ can be modified, whereas other Higgs
decay widths are intact. The decay width of H → γγ [68] is given by
2 v2 αem
H Γ(H → γγ) =
λHh A0 (τh+ ) + 4λHk A0 (τk++ )
3
64π mH –7–
JHEP09(2014)153
0.02
mH =125 GeV, mX =130 GeV
1 ´ 10-8
ΣpHpbL
5 ´ 10-9
1 ´ 10-9
5 ´ 10-10
1 ´ 10-10
1 ´ 10-11
0.00
0.02
0.04
0.06
0.08
0.10
ΛHX
Figure 4. The spin-independent cross section of dark matter scattering off proton, σp , as a function
of λHX (red solid line) and σp = 1.8 × 10−9 pb line above which is excluded by LUX (black dashed
line). We take mH = 125 GeV and mX = 130 GeV.
2
X
g2
+
Q2f Ncf A1/2 (τf ) ,
A1 (τW ) +
2τW
(2.14)
f =t,b
where τi = 4m2i /m2H (i = f, W, h+ , k ++ ). For the H → Zγ we adapted the formulas in
refs. [69, 70] for our model:
3
m2Z
G2F m2W αem m3H
1− 2
Γ(H → Zγ) =
64π 4
mH
f
2Nc Qf (T 3L − 2Qf xW )
2
λHh (c2W − 1)vH
f
A0 (τh+ , λh+ )
A1/2 (τf , λf ) + A1 (τW , λW ) +
×
cW
m2h+
2
2
λHk (c2W − 1)vH
A0 (τk++ , λk++ )
(2.15)
+
2
mk++
where. The loop functions are
A0 (τ, λ) = I1 (τ, λ),
A1/2 (τ, λ) = I1 (τ, λ) − I2 (τ, λ),
!
!
s2W τ2 + 1
s2W
2
A1 (τ, λ) = cW
− − 5 I1 (τ, λ) + 4 3 − 2
I2 (τ, λ) , (2.16)
τ
c2W
cW
where the function f is defined in (2.13) and
λ2 τ 2 f τ1 − f
I1 (τ, λ) =
2(τ − λ)2
1
λ
λτ 2 g τ1 − g
+
(τ − λ)2
–8–
1
λ
+
λτ
,
2(τ − λ)
JHEP09(2014)153
5 ´ 10-11
H b L m h + =150 GeV , m k ++ =500 GeV
H a L m h + =130 GeV , m k ++ =500 GeV
1
1
0.54
8
8
6
6
0.54
4
4
Λ Hk
Λ Hk
1.11
1.35
1.45
2
1.08
2
0
-2
-2
1
-3
-2
-1
0
0.91
1
2
1
3
-3
-2
Λ Hh
-1
0
1
0.92
2
3
Λ Hh
Figure 5. A contour plot for constant Γ(H → γγ)/Γ(H → γγ)SM (black solid lines) and Γ(H →
Zγ)/Γ(H → Zγ)SM (black dashed lines) in the (λHh , λHk ) plane. The shaded regions are disfavored
by (2.5) (blue) and by (2.7) (yellow). We set mh+ = 130 (150) GeV for the left (right) panel and
fixed mk++ = 500 GeV.
λτ f τ1 − f λ1
I2 (τ, λ) = −
,
2(τ − λ)
q
1
−1 √


, (τ ≥ 1)
 τ − 1 sin ( τ ) q
!
!
q
g(τ ) =
1+ 1− τ1
1
1


 2 1 − τ log 1−q1− 1 − iπ
(τ < 1).
(2.17)
τ
In figure 5, we show contour plots for constant Γ(H → γγ)/Γ(H → γγ)SM (black solid
lines) and Γ(H → Zγ)/Γ(H → Zγ)SM (black dashed lines) in the (λHh , λHk ) plane. For
this plot we set mh+ = 130 (150) GeV for the left (right) panel and fixed mk++ = 500 GeV.
The shaded regions are disfavored by (2.5) (blue) and by (2.7) (yellow). The ratios depend
basically only on the coupling constants λHh and λHk as well as the masses mh+ and mk++ .
And the ratios are not necessarily correlated with the hσviγγ which are controlled by λXh
and λXk . We can conclude that
0.54 . Γ(H → γγ)/Γ(H → γγ)SM . 1.45 (1.35)
0.91 . Γ(H → Zγ)/Γ(H → Zγ)SM . 1.11 (1.08)
for the left (right) panel. That is, the H → γγ channel can be enhanced (reduced) significantly, whereas the H → Zγ channel can change only upto ∼ 10%.
3
Spontaneously broken U(1)B−L model
As we have seen in the previous section, the simplest extension of Zee-Babu model to
incorporate dark matter with Z2 symmetry, although very predictive, has difficulty in fully
–9–
JHEP09(2014)153
0
explaining the Fermi-LAT gamma-line anomaly. In this section we consider a next minimal
model where we may solve the problem. We further extend the model by introducing
U(1)B−L symmetry and additional complex scalar ϕ to break the global symmetry [49, 50].
Then the model Lagrangian (2.3) is modified as
−LHiggs+DM = −µ2H H † H + µ2X X ∗ X + µ2h h+ h− + µ2k k ++ k −− − µ2ϕ ϕ∗ ϕ
+(µϕX ϕXX + h.c.)
+(λµ ϕh− h− k ++ + h.c.)
+λH (H † H)2 + λϕ (ϕ∗ ϕ)2 + λX (X ∗ X)2 + λh (h+ h− )2 + λk (k ++ k −− )2
+λϕX ϕ∗ ϕX ∗ X + λϕh ϕ∗ ϕh+ h− + λϕk ϕ∗ ϕk ++ k −−
+λXh X ∗ Xh+ h− + λXk X ∗ Xk ++ k −− + λhk h+ h− k ++ k −− ,
(3.1)
where we also replaced the real scalar dark matter X in (2.3) with the complex scalar field.
The charge assignments of scalar fields are given as follows:
H
U(1)Y
U(1)B−L
1
2
0
h+
1
2
k ++
2
2
ϕ
0
2
X
0
−1
We note that the soft lepton number breaking term µhk h+ h+ k −− in (2.3) is now
replaced by the B − L preserving λµ ϕh+ h+ k −− term. The U(1)B−L symmetry is spontaneously broken after ϕ obtains vacuum expectation value (vev). In [46], it was shown
that µhk < O(1) mh+ to make the scalar potential stable. In this U(1)B−L model, this can
be always guaranteed by taking small λµ , since µhk = λµ vϕ even for very large vϕ . The
term µXϕ XXϕ leaves Z2 symmetry unbroken after U(1)B−L symmetry breaking. Under
the remnant Z2 symmetry, X is odd while all others are even. It appears that the theory
is reduced to Z2 model in (2.3) when ϕ is decoupled from the theory. But we will see that
this is not the case and the effect of ϕ is not easily decoupled.
After H and ϕ fields get vev’s, in the unitary gauge we can write
!
0
1
H = √1
, ϕ = √ (vϕ + φ)eiα/vϕ ,
(3.2)
(v + h)
2
2 H
where α is the Goldstone boson associated with the spontaneous breaking of global
U(1)B−L . For convenience we also rotated the field X
X → Xe−iα/2vϕ
(3.3)
so that the Goldstone boson does not appear in the µϕX ϕXX term. Then the Goldstone
boson interacts with X via the usual derivative coupling coming from the kinetic term of
X-field.
The neutral scalar fields h and φ can mix with each other to give the mass eigenstates
Hi (i = 1, 2) by rotating
!
!
!
h
c H sH
H1
,
(3.4)
=
H2
φ
−sH cH
– 10 –
JHEP09(2014)153
+λHϕ H † Hϕ∗ ϕ + λHX H † HX ∗ X + λHh H † Hh+ h− + λHk H † Hk ++ k −−
where cH ≡ cos αH , sH ≡ sin αH , with αH mixing angle, and we take H1 as the SMlike “Higgs” field. Then mass matrix can be written in terms of mass eigenvalues m2i of
Hi (i = 1, 2):
!
!
2
m21 c2H + m22 s2H (m22 − m21 )cH sH
2λH vH
λHϕ vH vϕ
(3.5)
=
(m22 − m21 )cH sH m21 s2H + m22 c2H
λHϕ vH vϕ 2λϕ vϕ2
where αH is obtained from the relation
λHϕ vH vϕ
2 .
λϕ vϕ2 − λH vH
(3.6)
As will be discussed later, we may need parameter region where vϕ (∼ 106 GeV) is very large
2 ).
but m2 is at electroweak scale. From (3.5), we get λϕ ≈ m22 /(2vϕ2 ) and λH ≈ m21 /(2vH
The vacuum stability condition similar to (2.7) gives a constraint on λHϕ :
λHϕ .
m1 m2
,
vH vϕ
(3.7)
which is much stronger than the current bound from invisible Higgs decay width at the
LHC.
There is a mass splitting between the real and imaginary part of X:
X=
XR + iXI
√
.
2
(3.8)
In the scalar potential we have 22 parameters in total. We can trade some of those parameters for masses,
1 2
2
(m + m2I − λHX vH
− λϕX vϕ2 ),
2 R
m2 − m2I
µϕX = R√
,
2 2vϕ
1
1
2
µ2h = m2h+ − λHh vH
− λϕh vϕ2 ,
2
2
1
1
2
2
2
µk = mk++ − λHk vH − λϕk vϕ2 ,
2
2
µ2X =
(3.9)
(3.10)
(3.11)
(3.12)
where mR(I) is the mass of XR(I) . For simplicity we take XR as the dark matter candidate
from now on. We can also express λH , λϕ , λHϕ in terms of masses m2i (i = 1, 2) and mixing
angle αH , then we take the 22 free parameters as
vH (≃ 246 GeV),
vϕ ,
m1 (≃ 125 GeV),
m2 ,
mR ,
mI ,
mh+ ,
mk++ ,
λµ ,
λh ,
λk ,
λX ,
λHX ,
λϕX ,
λHh ,
λHk ,
λϕh ,
λϕk ,
αH ,
λXh ,
λXk ,
λhk ,
where two values, vH and m1 , have been measured as written in the parentheses.
– 11 –
(3.13)
JHEP09(2014)153
tan 2αH =
3.1
XR XR → γγ and Fermi-LAT 130 GeV γ-ray excess in U(1)B−L model
where Qi is electric charge of i(= h+ , k ++ ), τi = 4m2i /s and Γφ is total decay width of φ.
2 /4) ≈ 4m2 . When α = 0,
Since vrel ≈ 10−3 ≪ 1, we can approximate s = 4m2R /(1 − vrel
H
R
the H2 (= φ) can decay into two Goldstone bosons (α) or into two photons with partial
decay width
Γ(φ → αα) =
m3φ
,
32πvϕ2
v
u
√
2
4m2R(I)
(± 2µϕX + λϕX vϕ ) u
t
1−
,
Γ(φ → XR(I) XR(I) ) =
32πmφ
m2φ
v
u
2
u
4m2h+ (k++ )
(λ
v
)
ϕ
ϕh(k)
+ − ++ −−
t
,
Γ(φ → h h (k k )) =
1−
16πmφ
m2φ
2
X
2
2
αem vϕ
2
.
Q
λ
[1
−
τ
f
(τ
)]
Γ(φ → γγ) =
ϕi
i
i
i
64π 3 mφ (3.15)
(3.16)
(3.17)
(3.18)
i=h,k
Then the total decay width of φ is the sum:
Γφ = Γ(φ → αα) + Γ(φ → XR(I) XR(I) ) + Γ(φ → h+ h− (k ++ k −− )) + Γ(φ → γγ).(3.19)
As mentioned above, figure 6 shows the two enhancement mechanisms for XR XR → γγ:
the left panel for the φ-resonance and the right panel for the large vϕ . For these plots we set
the parameters: mR = 130, mI = 2000, mh+ = 300, mk++ = 500 (GeV), λϕh = λϕk = 0.1,
λϕX = λXh = λXk = 0.01, vϕ = 1000 (GeV) for the left plot and mφ = 600 (GeV) for
the right plot. We can obtain the large annihilation cross section required to explain
Fermi-LAT gamma-line data either near the resonance, mφ ≈ 2mR (left panel) or at
large vϕ (right panel). These behaviors can be understood easily from (3.14). In either
of these cases only the 1st term in (3.14) gives large enhancement. The slope on the
– 12 –
JHEP09(2014)153
In this section we will see that we can obtain dark matter annihilation cross section into
two photons, XR XR → γγ, large enough to explain the Fermi-LAT 130 GeV γ-line excess.
There are two mechanisms to enhance the annihilation cross section in this model: H2 resonance and large vϕ . In these cases, since the SM Higgs, H1 , contribution is small for
small mixing angle αH , we consider only the contribution of H2 assuming αH = 0 (or H2 =
φ). Allowing non-vanishing αH would only increase the allowed region of parameter space.
Then we obtain the annihilation cross section times relative velocity for XR XR → γγ,
√
2
αem ( 2µϕX + λϕX vϕ )vϕ X 2
σvrel (XR XR → γγ) =
Qi λϕi [1 − τi f (τi )]
32π 3 s s − m2φ + imφ Γφ
i=h,k
2
X
2
Qi λXi [1 − τi f (τi )] ,
(3.14)
+
i=h,k
0.1
ΣvHpbL
ΣvHpbL
0.1
0.001
10-5
10-5
10-7
0.001
10-7
100
150
200
300
500
700
1000
100
1000
mΦ HGeVL
104
105
106
vj HGeVL
0
-1
log10 ÈΛjX È
-2
-3
-4
-5
-6
-7
2
3
4
5
6
7
log10 Hvj GeVL
Figure 7. Contour plot of σv(XR XR → γγ) = 0.04 (pb) in (vϕ , λϕX )-plane. The red lines represent
the φ-resonance solution and the blue lines represent the large vϕ solution. The solid (dashed) lines
are for positive (negative) λϕX . See the text for the parameters chosen for this plot.
right of the resonance peak (the left panel of figure 6) is steeper than that on the left
because, when mφ > 260 GeV, new annihilation channel φ → XR XR opens and the decay
width of φ increases leading to decreasing the annihilation cross section. In the right
panel of figure 6, the dip near vϕ ≈ 104 GeV occurs because there is cancellation between
√
2µϕX = (m2R − m2I )/(2vϕ ) and λϕX vϕ terms for positive λϕX .
– 13 –
JHEP09(2014)153
Figure 6. Plots of σv(XR XR → γγ) for αH = 0 as a function of mφ (= m2 ) (left panel) and vϕ
(right panel). We set mR = 130, mI = 2000, mh+ = 300, mk++ = 500 (GeV), λϕh = λϕk = 0.1,
λϕX = λXh = λXk = 0.01, vϕ = 1000 (GeV) for the left panel and mφ = 600 (GeV) for the
right panel. The horizontal red line represent σv(XR XR → γγ) = 0.04 (pb) which can explain the
Fermi-LAT gamma-line signal.
3.2
Relic density in U(1)B−L model
Now we need to check whether the large enhancement in XR XR → γγ signal is consistent
with the observed relic density ΩDM h2 = 0.1199 ± 0.0027. To obtain the current relic
density the DM annihilation cross section at decoupling time should be approximately
(assuming S-wave annihilation)
hσvith ≈ 3 × 10−26 cm3 /s ≈ 1 pb,
(3.20)
from (1.1). The major difference between the Z2 model and the U(1)B−L model is
that the latter model has additional annihilation channel, i.e., XR XR → αα and φexchange s-channel diagrams compared with the former one. The S-wave contribution
to σv(XR XR → αα) is shown in the appendix. The Goldstone boson mode becomes dominant especially when vϕ is not very large [50], i.e. vϕ . 103 GeV. And it makes the dark
matter phenomenology very different from the one without it. For example, in Z2 model
we need the annihilation channel XX → h+ h− (k ++ k −− ) large enough to obtain the current relic density. In U(1)B−L model, however, the annihilation into Goldstone bosons are
sometimes large enough to explain the relic density.2
To see the relevant parameter space satisfying both the Fermi-LAT 130 GeV gammaline anomaly and the correct relic density, we consider the φ-resonance and large vϕ cases
discussed above separately. Figure 8 shows contours of σv(XR XR → γγ) = 0.04 (pb)
(solid line) and ΩXR h2 ≈ 0.12 (dashed line) for λϕX > 0 when the resonance condition
mφ = 2mR is satisfied. The parameters are chosen as mφ = 2mR = 260 GeV, mI = mh+ =
mk++ = 1 TeV, λϕh = λϕk = λXh = λXk = 0.01. We can see there are intersection points
of the two lines where both Fermi-LAT anomaly and the relic density can be explained.
For the parameters we have chosen the contribution of XR XR → αα to the relic density is
almost 100%. This implies there is wide region of allowed parameter space satisfying both
2
The dark sector can be in thermal equilibrium with the SM plasma in the early universe even with very
small mixing αH ∼ 10−8 [51]. And our analysis with αH = 0 can be thought of as a good approximation
of more realistic case of non-zero but small αH .
– 14 –
JHEP09(2014)153
The two mechanisms can also be seen in figure 7. This figure shows a contour plot
of σv(XR XR → γγ) = 0.04 (pb) in (vϕ , λϕX )-plane. We set mR = 130, mI = 1000,
mh+ = 1000, mk++ = 1000, mφ = 260 (GeV), λϕh = λϕk = λXh = λXk = 0.01 for red
lines (φ-resonance). And we take mI = 1000, mh+ = 300, mk++ = 500, mφ = 600 (GeV),
λϕh = λϕk = 0.1, λXh = λXk = 0.01 for blue line (large vϕ ). The red (blue) lines represent
the φ-resonance (large vϕ ) solution for Fermi-LAT anomaly. In the φ-resonance region, for
√
the negative (positive) λϕX the two values 2µϕX = (m2R − m2I )/(2vϕ ) and λϕX vϕ which
appear in the 1st term of (3.14) have the same (opposite) sign and their contributions are
constructive (destructive). As a result for positive λϕX (solid red line), there is cancellation
between the two terms, and larger value of vϕ is required for a given λϕX . For large vϕ case,
the result does not depend on the sign of λϕX because the λϕX vϕ term dominates. And the
solid and dashed blue lines overlap each other in figure 7. For λϕX larger than about 0.1 the
decay width Γ(φ → XR XR ) becomes too large to enhance the annihilation cross section.
0
-1
log10 ΛjX
-2
-3
-4
-6
-7
2
3
4
5
6
7
log10 Hvj GeVL
Figure 8. Contour plots of σv(XR XR → γγ) = 0.04 pb (solid red line) and ΩDM h2 = 0.1199
(dashed black line) in the (vϕ , λϕX )-plane for λϕX > 0.
We take the parameters,
2mR = mφ = 260 GeV. See the text for other parameters. The region enclosed by the
dashed lines gives ΩDM h2 > 0.12.
observables, since other annihilation channels XX → h+ h− (k ++ k −− ) are also available
when they are kinematically allowed. Typically TeV scalar vϕ gives too large XR XR → αα
annihilation cross section resulting in too small relic density. For the positive λϕX case,
however, there is also cancellation between terms in σv(XR XR → αα) as in σv(XR XR →
γγ). Both cancellations are effective when the condition, λϕX vϕ2 = (m2I −m2R )/2, is satisfied.
This explains the intersection point occurs on the diagonal straight line determined by the
above condition. This allows large relic density even near TeV vϕ .
Figure 9 shows the same contours for λϕX < 0. In this case as we have seen in figure 7
that TeV scale vϕ can explain Fermi-LAT gamma-line. However this value of vϕ gives
too large DM annihilation cross section at the decoupling time (when XR XR → αα is
dominant) and too small relic density. So somehow we need to “decouple” the XR XR →
γγ so that we need larger vϕ . We can do it, for example, by assuming h+ (k ++ ) are
very heavy: mh+ = mk++ = 5 TeV. If we also reduce the mass difference mI − mR , we
√
get smaller 2µϕX = −(m2I − m2R )/(2vϕ ). Then we have simultaneous solution both for
σv(XR XR → γγ) ≈ 0.04 pb and ΩXR h2 ≈ 0.12 as can be seen in figure 9. The two
lines meet at rather large vϕ (∼ 105 GeV) as expected. For other parameters we chose3
mφ ≈ 2mR = 260 GeV, mI = 200 GeV, λϕh = λϕk = λXh = λXk = 0.01. The pattern
of the relic density contour requires some explanation. The annihilation cross section for
3
To avoid fine tuning we allowed small off-resonance condition of the size of Γφ ∼ 1 keV, i.e., we set
mφ = 260.000001 GeV.
– 15 –
JHEP09(2014)153
-5
0
log10 H-ΛjXL
-2
-4
-6
-10
3
4
5
6
7
8
log10 Hvj GeVL
Figure 9. The same plot with figure 8 for λϕX < 0. We also take the φ-resonance condition,
2mR = mφ = 260 GeV. See the text for other parameters. The region to the right of the dashed
line gives ΩDM h2 > 0.12.
XR XR → αα at the resonance can be approximated from (B.1) as
√
( 2µϕX + λϕX vϕ )2
σv(XR XR → αα) ≈
,
16πvϕ2 Γ2φ
(3.21)
where Γφ is the total decay width of φ. The non-vanishing partial decay widths of φ for
the parameters we chose are Γ(φ → αα), Γ(φ → XR XR ) and Γ(φ → γγ). In figure 10
they are plotted as a function of vϕ for λϕX = −10−7 . On the vertical part of the relic
density contour in figure 9 near vϕ ∼ 104.4 GeV, the Γ(φ → αα) dominates and also
√
2µϕX ≫ λϕX vϕ . For this vϕ we approximately get
σv(XR XR → αα) ∼
16πm4I
,
m6φ
(3.22)
which is independent of λϕX . Around vϕ ∼ 107.2 GeV, Γ(φ → XR XR ) and Γ(φ → γγ)
√
dominate despite high phase space suppression in φ → XR XR , and 2µϕX ≪ λϕX vϕ .
As λϕX increases, Γ(φ → XR XR ) becomes more important than Γ(φ → γγ) as can be
seen from (3.15) and (3.16). The (almost) vertical part for this vϕ region is due to partial
√
cancellation of the factor ( 2µϕX +λϕX vϕ )2 in the numerator of (3.21) and the same factor
in the cross term of Γ(φ → XR XR ) and Γ(φ → γγ) in the denominator. As λϕX grows
even larger, only Γ(φ → XR XR ) term dominates and σv(XR XR → αα) ∝ 1/λ2ϕX vϕ4 , which
gives the slanted part of the contour line.
We can also obtain simultaneous solutions when φ is off-resonance using large vϕ .
Figure 11 shows an example of this case. In this case, if we have only XR XR → ααchannel for relic density, the resulting ΩXR h2 is too large for vϕ & 1 TeV. To get the
– 16 –
JHEP09(2014)153
-8
Partial Width HGeVL
10-4
10-6
10-8
10-10
10-12
4
5
6
7
8
Figure 10. Plots of Γ(φ → αα) (solid blue line), Γ(φ → XR XR ) (dashed red line) and Γ(φ → γγ)
(dotted green line) as a function of vϕ for the parameters used in figure 9. We fixed λϕX = −10−7 .
0
-1
log10 ΛjX
-2
-3
-4
-5
-6
-7
2
3
4
5
6
7
log10 Hvj GeVL
Figure 11. The same plot with figure 8 corresponding to large vϕ solution. See the text for the
parameters used in this figure.
correct relic density by increasing the DM pair annihilation cross section at freeze-out we
allowed XR XR → h+ h− channel. Then we can get a solution as can be seen in figure 11.
The region enclosed by two dashed lines over-closes the universe. For this plot, we chose
mR = 130 GeV, mφ = mI = 1 TeV, mh+ = 150 GeV, mk++ = 500 GeV, λϕh = 0.001, and
λϕh = λϕk = λXh = λXk = 0.01. Note that we take λϕh = 0.001 so that the solid red
line representing σv(XR XR → γγ) = 0.04 pb and dashed line representing Ωh2 = 0.1199
overlap with each other. To show that this choice of λ’s is possible in general, we take a
point on the overlapped lines, e.g., vϕ = 106.43 GeV and λϕX = 0.001. Then we can get
λϕh = 0.001, λϕh = 0.01 as a solution from figure 12.
– 17 –
JHEP09(2014)153
log10 Hvj GeVL
0
-1
log10 Λjk
-2
-3
-4
-6
-7
-7
-6
-5
-4
-3
-2
-1
0
log10 Λjh
Figure 12. Contours of σv(XR XR → γγ) = 0.04 pb (solid red line) and Ωh2 = 0.1199 (dashed
blue line).
When there is no mixing between φ and h the decay width H1 → γγ is the same with
that of the SM. This means that we can enhance the XR XR → γγ without affecting the
SM H1 → γγ rate. When the mixing angle αH is non-vanishing the h+ and/or k ++ can
contribute to H1 → γγ through one-loop process. Since this effect was already discussed
in section 2.4, we do not discuss it further.
3.3
Other cosmological implications of U(1)B−L model: ∆Neff , topological
defects, self-interacting dark matter
Weinberg [52] showed that Goldstone bosons can play the role of dark radiation and
contribute to the effective number of neutrinos Neff . If Goldstone bosons go out of
equilibrium when the temperature is above the mass of muons but below that of all other
SM particles, we get ∆Neff = 0.39. The condition for this to happen can be roughly
estimated by setting the collision rate nhσvi(αα → µµ) is equal to the Hubble expansion
rate H ∼ T 2 /mpl . The αα → µµ is dominated by the s-channel scalar exchange diagram.
It is mediated by an operator
−
λHϕ mµ
∂µ α∂ µ αµµ,
m2φ m2H
(3.23)
which is generated by terms −1/vϕ φ∂µ α∂ µ α, −λHϕ vϕ vH φh and mµ /vH µµ in (3.1). Since
n ∼ T 3 and a derivative yields a factor T in thermal averaging, nhσvi ∼ H gives
λ2Hϕ m2µ
m4φ m4H
T7 ∼
– 18 –
T2
.
Mpl
(3.24)
JHEP09(2014)153
-5
At T ∼ mµ , the condition becomes
λ2Hϕ m7µ Mpl
m4φ m4H
∼ 1.
(3.25)
where mpl ∼ 1019 GeV is Planck mass and µ ∼ vϕ2 ln(vϕ d) width d the characteristic
distance between strings. The constraint µ/m2pl . 10−6 [61] is easily satisfied for the values
of vϕ . 107 GeV taken in our scenario.
After U(1)B−L symmetry breaking, our model has remnant discrete Z2 symmetry. If
this Z2 symmetry is spontaneously broken, domain walls can be formed and dominate the
energy density of the universe, causing domain wall problem [62].
If inflation occurs after the U(1)B−L symmetry breaking, these topological defects
are diluted and do not make any problems. However, recent BICEP2 observation [3]
suggests that the inflation scale may be much higher than the U(1)B−L breaking scale.
And we cannot resort to inflation to solve the domain wall problem, if they are produced.
However, if we assume the Goldstone boson is massless, the potential along the α-direction
is flat (Z2 is exact), the domain walls are not produced, because the Z2 symmetry is
not spontaneously broken. In more general, the Goldstone bosons get masses from higher
dimensional operators suppressed by Planck mass which are generated by quantum gravity.
This can break Z2 symmetry explicitly and there is no domain wall problem, although
some higher dimensional operators should be suppressed to guarantee the longevity of
– 19 –
JHEP09(2014)153
After α being decoupled from thermal plasma, the energy from muon annihilation heats up
neutrinos but not α. This enhances the neutrino temperature relative to the temperature
of the Goldstone boson. From the entropy conservation before and after muon annihilation,
we get Tν /Tα = (57/43)1/3 . The ∆Neff is equal to the energy fraction of a Goldstone
boson relative to a single neutrino: ∆Neff = Tα4 /(7/4Tν4 ) = 4/7(43/57)4/3 = 0.39. Actually
WMAP9 and ground-based observations [53–55] give Neff = 3.89 ± 0.67 and Planck, the
WMAP9 polarization and ground-based observations [56–59] give Neff = 3.36 ± 0.34, both
at the 68% confidence leve, suggesting possible deviation from the SM prediction although
the errors are large. For example, with λHϕ = −0.005(−0.0001) and mH = 125 GeV,
the dark scalar with mass 500 MeV (70 MeV) can satisfy the condition. With this light
dark scalar, obviously the resonance solution for Fermi-LAT gamma line is not applicable.
From (3.5) we can see with λHϕ = −0.0001 and m2 = 70 MeV, if we take sH = 0.3 which
is consistent with Higgs invisible decay width at the LHC, we get λϕ = 2 × 10−8 and
vϕ = 190 TeV, which is consistent with both (2.7) and figure 11 for the Fermi-LAT gamma
line. As analyzed in section 3.2, we can easily find the dark sector parameters to explain
the current relic abundance.
At the early universe when the U(1)B−L symmetry is broken, cosmic strings can be
produced. Although the energy per unit length of a long straight string from global symmetry breaking diverges logarithmically, the energy per length of two anti-parallel strings
is finite [60]. The contribution of cosmic strings to the total energy budget of the universe
is given by [60]
ΩS ∼ µ/m2pl ,
(3.26)
3.4
Implications for neutrino physics in both Z2 and U(1)B−L model
Since the new particles X (in both Z2 and U(1)B−L model) and ϕ (in U(1)B−L model)
beyond the Zee-Babu scalars h+ , k ++ do not couple directly to lepton doublets, the
neutrino phenomenology is the same with the original Zee-Babu model and no further
constraints are imposed by X or ϕ at least in the tree level. It is well known the Zee-Babu
model is strongly constrained by charged lepton flavor violating (LFV) processes such as
µ → eγ or τ → µµµ [47]. The µ-parameter is constrained to be less than about 500 GeV
to make the scalar potential stable [46].
The most recent constraints on Zee-Babu model were studied in [48]. In the analysis
they used the updated values of θ13 and µ → eγ and the updated LHC results. They found
that the neutrino oscillation data and low energy experiments are compatible with masses
of the extra charged scalar mass bounds from the LHC.
4
Conclusions
We have considered two scenarios which minimally extended the ‘Zee-Babu model’ [44–
46]. In the first scenario we introduced a real scalar dark matter X with Z2 symmetry:
X → −X. If the scalar dark matter X has a mass around 130 GeV, the annihilation cross
section, hσvi(XX → γγ), can be enhanced by the contribution of the singly- and/or doublycharged Zee-Babu scalars. If we also want to explain the dark matter relic abundance,
however, we get at most hσviγγ ≈ 0.2 × 10−27 cm3 /s, which is about factor 6 smaller than
the required value to explain the Fermi-LAT gamma-ray line signal.
We have shown that the present constraint on the couplings λXk and λXh which
mix the dark matter and charged Higgs is not so strong and they can enhance the
– 20 –
JHEP09(2014)153
dark matters. Our scenario is different from the axion case where the axion masses are
induced by QCD instantons making the axion potential have discrete symmetry that is
broken spontaneously.
The self-interacting dark matters [5–10] with long-range force have received much
interest because they can solve core/cusp problem [4] and “too big to fail” problem [11].
The current bound on self-interacting dark matters is given by
σT . 35cm2 /g,
(3.27)
mDM v=10km
R
where σT = dΩ(1 − cos θ)dσ/dΩ is the transfer cross section. If the above cross section
is close to the bound, the dark matters can solve the problems. The Goldstone boson
which is assumed to be massless or very light couples to the dark matter with interaction
suppressed by vϕ as can be seen from the rotation in (3.3). Although the Goldstone boson
could be thought to mediate long-range force, actually the force scales not as 1/r but as
1/r3 due to its pseudo-scalar nature [12], and cannot contribute to the self-interactions
of dark matters much. However, if the scalar φ is light (sub-GeV scale) it can mediate
long-range Yukawa interactions making our dark matter self-interacting. It is interesting
to see that light φ can enhance both ∆Neff and the self-interactions.
Acknowledgments
We thank Wan-Il Park for useful discussions. This work is partly supported by NRF
Research Grant 2012R1A2A1A01006053 (PK, SB).
A
One-loop β functions of the quartic couplings
Here, we give the renormalization group equation and the one-loop β functions of the
quartic couplings:
dλi
= β λi ,
(A.1)
d ln Q
with
β λH
β λh
o
λ2HX
3n 4
1
2
2 2
4
2
2
2
2g
+
(g
+
g
)
−
6y
+
24λ
+
λ
+
λ
+
=
2
2
1
t
H
Hh
Hk
16π 2
2
8
3
2
2
2
− 4λH
(3g + g1 ) − 3yt
, (A.2)
4 2
λ2Xh
1
4
2
2
2
2
=
+ 6g1 − 12λh g1 ,
16λh + 2λHh + λhk +
(A.3)
16π 2
2
– 21 –
JHEP09(2014)153
annihilation cross section of XX → γγ large enough to accommodate the recent hint. On
the other hand the couplings which involve the SM Higgs H are strongly constrained by
the theoretical considerations in the Higgs potential and the observations of dark matter
relic density and dark matter direct detections. The upper bound on the λHX coupling
is about 0.06 which comes from the dark matter direct detection experiments. For the
λHh , λHk which mix the SM Higgs and the new charged Higgs, the theoretical bound
becomes more important. If we require the absolute stability of the dark matter by the
Z2 symmetry X → −X and the absence of charge breaking, we get the upper bound
of λHh , λHk to be about 0.7 for the charged Higgs mass around 150 GeV. To evade the
unbounded-from-below Higgs potential we need to have λHh , λHk & −1.6. With these
constraints the B(H → γ(Z)γ) can be enhanced up to 1.5 (1.1) or suppressed down to
0.5 (0.9) with respect to that in the SM. The neutrino sector cannot be described by the
Zee-Babu model only, and there should be additional contributions to the neutrino masses
and mixings such as dimension-5 Weinberg operator from type-I seesaw mechanism.
In the second scenario, we introduced two complex scalar fields X and ϕ with global
U(1)B−L symmetry. After ϕ gets vev, vϕ , the U(1)B−L symmetry is broken down to Z2
symmetry. The lighter component of X, which we take to be the real part, XR , is stable
due to the remnant Z2 symmetry and can be a dark matter candidate. Even in the extreme
case where we do not consider the mixing of the dark scalar and the standard model Higgs
scalar (αH = 0), we showed that the dark matter relic abundance and the Fermi-LAT
gamma-ray line signal can be accommodated in two parameter regions: resonance region
(mφ = 2mR ) and large vϕ (∼ 106 − 107 GeV) region. Since there is no mixing, there is no
correlation with H → γγ and direct detection scattering of dark matter off the proton. In
addition the neutrino sector need not be modified contrary to the first scenario.
β λk =
β λX =
βλHh =
βλHk =
βλHX =
βλXh =
βλXk =
B
The annihilation cross section of XR XR → αα
The annihilation cross section of XR XR → αα is obtained as:
h
i2
√
m2R 4vϕ ( 2µϕX + λϕX vϕ )(m2I + m2R ) + (m2I − m2R )(4m2R − m2φ )
σv(XR XR → αα) =
64πvϕ4 (m2I + m2R )2 (4m2R − m2φ )2
+O(v 2 ),
(B.1)
where (4m2R − m2φ )2 should be replaced by (4m2R − m2φ )2 + Γ2φ m2φ in the denominator when
2mR ≈ mφ .
Open Access. This article is distributed under the terms of the Creative Commons
Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
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